We each took notes individually on things we noticed about the lawn mower problem and questions we had, which we then shared at our tables. After our tables had discussed our noticings and wonderings, Tyler brought us back together to share with the whole group:

With the very first noticing, we all agreed that though many questions presented themselves, there was no actual question to tell us what to do with the situation involving the three lawnmowers.

Tyler’s facilitation of the conversation seemed focused on two things – having participants explain their ideas precisely and making sure everyone really understood the situation. His most go-to questions were “Can you say a little more about that?” and “What do the rest of you think?”

After we had reached some consensus on the meaning of various details – like the fact that in the sentence “One lawn keeper believes they’ll only need…”, the third-person plural (they’ll) is referring to the third lawn keeper and “Each lawn keeper knows they can cut…”, the “they” is referring to each individual lawn keeper.

Tyler then told us that we had actually come up with the question he wanted us to work on:

How many square yards is the lawn?

He then asked us to work towards answering that question on our own. We work individually for about 10 minutes, then started to talk with each other at our tables.

Solution Methods

Ramon brought us back together to discuss different solutions in the following order (we didn’t have time to discuss it, but it seemed like he and Tyler had a rationale for the order we looked at each method):

First we explored Avril’s strategy:

Avril started with a visual representing the entire field with each of the three different sections. She used the percentages, reasoning that 58% of the field would be mowed by the first two, which would mean that the 700 square yards represents what’s left – or 42% of the field. Avril solved for the missing whole by setting up a proportion. She wasn’t sure about her answer, so she reasoned further that if 700 was 42% then 1400 would be 84%. That helped her feel more comfortable with her answer.

Next up we looked at Leo and Theresa’s method:

They treated the 25% as a fraction and found that 7/12 of the field is mowed after the first two lawn keepers complete their section. They subtracted that piece from the whole field and realized that the final lawn keeper’s section of 700 sq yards was equivalent to the remaining 5/12 of the field. They set up a proportion and solved it through cross-multiplication. Ramon asked some nice questions to get us to make connections between the different methods and representations – like “Where does this 1-7/12=5/12 show up in Avril’s thinking?”

Then Mark shared his method:

Mark also started with fractions converting 25% to 1/4 and adding it to 1/3. That meant 7/12 of the field is covered (Avril’s 58%) and 5/12 remains. Instead of a proportion, Mark used a visual to show the situation. He drew a rectangle broken into twelfths and shaded in 5 of them. That represents the 700 sq yards. Mark’s reasoned that 700/5 would be equivalent to the area of the whole field divided by 12. If 5 pieces equal 700, then each individual piece would equal 140. He then multiplied 12 pieces by 140.

Finally, Ruben shared his method:

There was a lot of appreciation for the apparent simplicity of Ruben’s algebraic solution, but then there was some confusion about how he removed the fractions from the first equation he set up. A few folks noted that everything Ruben had done from the 4x + 3X + 700(12) = 12x on down was straightforward procedure in terms of solving for x, but that the initial equation he set up and the way he eliminated the fractions are much harder to teach because they require both an understanding of the situation, and of a way to express it algebraically. It is challenging because there are several equations he could have used to model the situation and different methods he could have used to begin to manipulate it.

Ramon shared one to demonstrate this point:

Some Big Teaching Questions and Observations

Some questions that came out of our discussion:

Should we prioritize the teaching of algebra over other problem-solving methods? What if a visual method works better for some students – should we allow students to process in a way that makes sense for them?

There is more than one way to skin a cat. Multiple methods of problem-solving is great but is there one best solution method for each problem. Should I show my “easy” way before looking at student methods, or vice-versa?

You can make really interesting connections between all four of these different solution methods. Could you use the methods that come from students to build to some of these other methods?

Is it possible to teach all topics in logical order, so that students understand least common denominator before manipulating algebraic equations? If not, how do we help students with fear of math, so they aren’t lost in the deep end?

Some opinions/observations that came out of our discussion:

Informal, individual math language can be as rigorous as algebra if every step and the thinking is shown without any gaps

I can do the math once I figure out how to set up the problem. The math is easy but how to set up the problem… how to make sense of a situation in a mathematical way?

Setting up an equation to solve for x in an algebraic equation is a creative act. There is more than one way to proceed.

Knowing how to do something is sometimes only possible if you know you can do it (Example: multiplying both sides by the least common denominator of fractions in the equation or multiplying both sides of an equation by the reciprocal of x)

Students should be familiar with the language of algebra because they will likely encounter “algebraic colloquialisms”

We can think of algebra as a language – Ruben’s fluency was demonstrated by his ability to express a real-life situation as an algebraic equation

Another Question

Tyler and Ramon had planned for us to answer a second question, but the methods and discussion of the area question was rich and we ran out of time. The second question (also taken from our wonderings) was “How long would it take to complete each section?”

Here was one approach:

Reasoning: Each can do it in 6 hours if they are working alone. That means the first guy who is mowing one-third of the lawn will finish in 2 hours. The guy who is doing twenty-five percent will finish in an hour and a half. That is 3 and a half hours of work. If the whole job would take the last guy six hours, then it would take him 2 and a half hours to do the remaining 700 square yards.

Annie Fetter from the Math Forum on asking Students “I Notice… I Wonder…”

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